Fundamental Graphs for Graph Drawing

Just like with standard integrals, in graph drawing there are a number of basic graphs that you should absolutely know how to draw.

Linear

$y = p$

For any $x$, we produce a constant $y$-value, $p$. In other words, it’s a horizontal straight line.

y = p
p = 2

$x = q$

What points in the 2D plane have an $x$-coordinate of $q$? It’s all the points along the vertical line intersecting the $x$-axis at $q$.

It’s identical to $y = p$, only in a different orientation. Just 2 different ways of thinking about the same concept.

x = q
q = 3

$y = x$

For every $x$-value, we output an equivalent $y$-value. This looks like a straight line at $45 \degree$ above the positive $x$-axis.

y = x

Polynomials

$x^2$

Famously known as a “happy parabola”, symmetrical in the $y$-axis.

y = x^2

$x^3$

y = x^3

$x^p$

For all $p \in \mathbb{Z}^{+}$, $x^p$ will always intersect the origin (because $0^p = 0$ as long as $p > 0$).

x^p

/slider{ min: 0.1 } :: p = 2

$x^{2p}$

All even positive powers of $x$ produce even graphs. This is because an even number of multiplications cancels out pairs of $-$ signs.

f\left( x \right) = x^{2p}
f\left( -x \right)

/slider{ min: 2, step: 2 } :: p = 4

$x^{2p-1}$

All odd positive powers of $x$ produce odd graphs. This is because one $-$ sign will be left unpaired and hence not cancelled out through the multiplications.

f\left( x \right) = x^{2p-1}
-f\left( -x \right)

/slider{ min: 1, step: 2 } :: p = 3

$(x-p)(x-q)$

Only when $x = p$ or $x = q$ does the output $y$ becomes $0$. That means the graph intersects the $x$-axis twice, at $(p, 0)$ and $(q, p)$.

This is a (positive) quadratic with roots at $x = p, x = q$.

y = \left( x-p \right) \left( x-q \right)
/asympt :: x = p
/dashed :: x = q

p = -2
q = 3

$(x-p)(x-q)(x-r)(...)$

More generally, any factorised polynomial in this form has roots $p, q, r, ...$ since whenever $x$ is one of these values the whole polynomial collapses to $0$.

y = \left( x-p \right) \left( x-q \right) \left( x-r \right)
/dashed :: x = p
/dashed :: x = q
/dashed :: x = r

p = -3
q = 1
/hidden :: r = 2

Fractional Powers

$\sqrt{x}$

y = \sqrt{x}

$x^{1/p}$

y = x^{1/p}

/slider{ min: 1 } :: p = 3

Exponentials & Logarithms

$e^x$

y = e^x

$b^x$

y = b^x

/slider{ min: 1 } :: b = 2

$\ln(x)$

y = \ln\left( x \right)

$\log_b(x)$

y = \log_{b}\left( x \right)

/slider{ min: 2 } :: b = 3

Reciprocals

$1/x$

y = \frac{1}{x}

$1/x^2$

y = \frac{1}{x^2}

$1/x^p$

y = \frac{1}{x^p}

/slider{ min: 1 } :: p = 3

Miscellaneous

$|x|$

y = \left| x \right|

$\max(x, 0)$

Also known as the rectified linear unit^↗, this looks like a ramp, $0$ for all negative values.

y = \operatorname{max}\left( x,\ 0 \right)